Physical Review A

Extended Salecker-Wigner formula for optimal accuracy in reading a clock via a massive signal particle

Shoju Kudaka, Shuichi Matsumoto

Published Jul 11, 2007• 1 citation

Abstract

In order to acquire an extended Salecker-Wigner formula from which to derive the optimal accuracy in reading a clock with a massive particle as the signal, von Neumann's classical measurement is employed, by which simultaneously both position and momentum of the signal particle can be measured approximately. By an appropriate selection of wave function for the initial state of the composite system (a clock and a signal particle), the formula is derived accurately. Valid ranges of the running time of a clock with a given optimal accuracy are also given. The extended formula means that contrary to the Salecker-Wigner formula there exists the possibility of a higher accuracy of time measurement, even if the mass of the clock is very small.

DOI: 10.1103/physreva.76.012108

Related papers

Dependence of the Time-Reading Process of the Salecker–Wigner Quantum Clock on the Size of the Clock

Andor Frenkel

It is shown in the present note that the degree of the complexity of the time-reading process of the Salecker–Wigner clock depends on the size of the clock. This dependence leads to a relation between the size and the accuracy of the clock, and suggests a precise optimal value for the size in agreement with the order of magnitude value established by Salecker and Wigner.

Jan 1, 20152 citations

Quantum Limitations of the Measurement of Space-Time Distances

H. Salecker, Eugene Paul Wigner

This article deals with the limitations which the quantized nature of microscopic systems imposes on the possibility of measuring distances between space-time events. It is proposed to use only clocks for measuring space-time distances and to avoid the use of measuring rods which are essentially macrophysical objects. The accuracy of reading a clock with a given mass is considered and examples for microphysical clocks are given. It is shown that the mass of the clock, and the uncertainty (spread) of this quantity, exceed certain values which depend on the accuracy with which the time interval is to be measured, the magnitude of this time interval (the running time of the clock) and the size of the clock. The minimum mass uncertainty of the clock is given by Heisenberg's relation; the minimum mass itself is higher by the ratio of the running time and the accuracy.If the possibility of constructing states whose wave functions are Gaussian wave packets is admitted, the mass and the mass uncertainty of the clock differs only by logarithmic factors from the uncertainties which follow from general principles of quantum mechanics. The masses are much higher if the possibility of constructing arbitrary wave packets is not admitted.

Physical ReviewJan 15, 1958386 citations

Barrier interaction time and the Salecker-Wigner quantum clock: Wave-packet approach

Chang-Soo Park

The time-of-flight measurement approach of Peres based on the Salecker-Wigner quantum clock is applied to the one-dimensional scattering of a wave packet from a rectangular barrier. By directly evaluating the expectation value of the clock-time operator in the asymptotic states of wave packet long after the scattering process, we derive an average wave-packet clock time for the barrier interaction, which is expressed as an average of the stationary-state clock time over all possible initial scattering states of the wave packet. We show that the average wave-packet clock time is identical to the average dwell time of a wave packet.

Physical Review AJul 23, 200910 citations

Salecker-Wigner-Peres clock and double-barrier tunneling

Marcos Calçada, Jose Tadeu Lunardi, Luiz A. Manzoni

We reexamine the Salecker-Wigner-Peres clock formalism and show that it can be directly applied to the phenomenon of tunneling. Then we apply this formalism to the determination of the tunneling time of a nonrelativistic wave packet, sharply concentrated around a tunneling energy, incident on a symmetric double-barrier potential. In order to deepen the discussion about the generalized Hartman effect, we consider the case in which the clock runs only when the particle can be found inside the region between the barriers and show that, whenever the probability to find the particle in this region is non-negligible, the corresponding time (which in this case turns out to be a dwell time) increases with barrier spacing.

arXiv:0901.0508Physical Review AJan 22, 200923 citations

Salecker-Wigner-Peres clock, Feynman paths, and a tunneling time that should not exist

D. Sokolovski

The Salecker-Wigner-Peres (SWP) clock is often used to determine the duration a quantum particle is supposed to spend in a specified region of space $\mathrm{\ensuremath{\Omega}}$. By construction, the result is a real positive number, and the method seems to avoid the difficulty of introducing complex time parameters, which arises in the Feynman paths approach. However, it tells little about the particle's motion. We investigate this matter further, and show that the SWP clock, like any other Larmor clock, correlates the rotation of its angular momentum with the durations $\ensuremath{\tau}$, which the Feynman paths spend in $\mathrm{\ensuremath{\Omega}}$, thereby destroying interference between different durations. An inaccurate weakly coupled clock leaves the interference almost intact, and the need to resolve the resulting ``which way?'' problem is one of the main difficulties at the center of the ``tunnelling time'' controversy. In the absence of a probability distribution for the values of $\ensuremath{\tau}$, the SWP results are expressed in terms of moduli of the ``complex times,'' given by the weighted sums of the corresponding probability amplitudes. It is shown that overinterpretation of these results, by treating the SWP times as physical time intervals, leads to paradoxes and should be avoided. We also analyze various settings of the SWP clock, different calibration procedures, and the relation between the SWP results and the quantum dwell time. The cases of stationary tunneling and tunnel ionization are considered in some detail. Although our detailed analysis addresses only one particular definition of the duration of a tunneling process, it also points towards the impossibility of uniting various time parameters, which may occur in quantum theory, within the concept of a single tunnelling time.

arXiv:1703.01966Physical Review AAug 14, 201715 citations

Wigner function and the successive measurement of position and momentum

Pier A. Mello, M. Revzen

Wigner function is a ``quasidistribution'' that provides a representation of the state of a quantum mechanical system in the phase space of position and momentum. In this paper we find a relation between the Wigner function and appropriate measurements involving the system's position and momentum which generalize the von Neumann model of measurement. We introduce two probes coupled successively in time to projectors associated with the system's position and momentum. We show that one can relate the Wigner function to the Kirkwood joint quasidistribution of position and momentum, the latter, in turn, being a particular case of successive measurements. We first consider the case of a quantum mechanical system described in a continuous Hilbert space and then turn to the case of a discrete, finite-dimensional Hilbert space.

arXiv:1307.2877Physical Review AJan 13, 201411 citations